Applications of homotopy theory to 4D geometry, number theory, and physics

0406461杰克·莫拉瓦同伦理论在最近的四维几何、物理和数论工作中扮演着重要的角色：它是一种技术上强大的意识形态，揭示了表面上看似无关的主题之间的深层联系。Madsen，Tillmann，Weiss，Cohen等人最近在Riemann曲面的模空间的稳定上同调上同调方面的工作是基于弦物理思想提出的某些协边范畴的研究。这些结构有四个维度的类似物(在物理上具有相似的意义)，它们与哈彻、沃尔德豪森和其他人在80‘S和90’S中发展的伪同位素理论有惊人的联系。该理论以一种非平凡的方式涉及到整数的代数K-理论；而后者现在被认为在一个明显不相关的思想圈中扮演着重要的角色，这个思想圈将有理数域的绝对伽罗瓦群与代数几何中的动机理论联系起来(通过Deligne，Goncharov，Kontsevich和其他人的工作)。康内斯、克雷梅尔和其他人对物理学中经典的重整化理论的重新解释也涉及到这些想法。这一提议的核心概念之一是希望在微分拓扑学中找到代数几何学家混合状态动机的模拟，这将与整数的代数K-理论有关，正如该主题与Waldhausen的空间的代数K-理论有关。用不那么专业的术语：同伦理论为在平等的基础上研究数学对象及其变形提供了一套技术；实际上，它同样乐于研究变形等。这一提议最终涉及到两套植根于物理学的思想，一套来自现代弦理论，另一套来自更经典的重整化理论。前者与微分拓扑学有很深的联系，后者与代数几何和数论的最新发展有关。从表面上看，微分拓扑和算术代数几何相距甚远，但它们通过整数的代数K-理论联系在一起，这是同伦理论的基础。[因为这门学科如此系统地研究从一种理论到另一种理论的变形，它提供了一套非常方便的工具来将这些不同的主题联系起来。]这一建议提出了一种澄清这些联系的方法，其基础是关于空间的K理论而不是数字的想法，这是由Waldhausen最先提出的，最近得到了像Soul\‘e这样的算术几何工作者的进一步支持。这个计划将以一种概念上令人满意的方式将数学和物理中的这些重要的最新发展结合在一起。

DMS-0406461 Jack MoravaHomotopy theory plays an important role in recent work in four-dimensional geometry, physics, and number theory: it is a technically powerful ideology, which exposes deep connections among topics which may on the surface seem unrelated. Recent work of Madsen, Tillmann, Weiss, Cohen, and others on the stable cohomology of the moduli space of Riemann surfaces is based on the study of certain cobordism categories suggested by ideas from string physics. These constructions have four-dimensional analogs (of similar significance in physics), which display surprising connections to the pseudoisotopy theory developed in the 80's and 90's by Hatcher, Waldhausen, and others. That theory involves the algebraic K-theory of the integers in a nontrivial way; and the latter subject is now seen to play an important role in an apparently unrelated circle of ideas connecting the absolute Galois group of the rational number field to the theory of motives in algebraic geometry (through work of Deligne, Goncharov, Kontsevich, and others). These ideas are in turn involved in work of Connes, Kreimer, and others on a reinterpretation of the classical theory of renormalization in physics. One of the central notions of this proposal is the hope of finding in differential topology an analog of the algebraic geometers' mixed Tate motives, which would be related to the algebraic K-theory of the integers as that subject is to Waldhausen's algebraic K-theory of spaces.In less technical terms: homotopy theory provides a body of techniques for studying mathematical objects and their deformations on an equal footing; indeed, it is equally happy studying deformations of deformations, and so on. This proposal is concerned ultimately with two sets of ideas with roots in physics, one coming from modern string theory, the other from the more classical theory of renormalization. The former set of ideas has deep connections with differential topology, and the latter is related to recent developments in algebraic geometry and number theory. On the surface, differential topology and arithmetic algebraic geometry are far apart, but they are linked through the algebraic K-theory of the integers, which is at base a part of homotopy theory. [Because that subject deals so systematically with deformations of one theory into another, it provides a very convenient set of tools for relating such disparate subjects.] This proposal suggests a way of clarifying these linkages, based on ideas about the K-theory of spaces rather than numbers, which were first proposed by Waldhausen, and which have lately given further currency by workers in arithmetic geometry such as Soul\'e. This program would bring together these important recent developments in mathematics and physics in a conceptually satisfying way.